Study on spectral representation and eigenexpansion based on eigen-operators

The spectral representation and expansion based on eigen-operators were deeply studied.The relations between Green’s function of characteristic differential equation and Hermitian differential and integral operators were given.The inverse relations between Hermitian differential operator and Hermitian integral operator were also studied.The spectral representation of Hermitian differential operators was given.It was also shown that the S-L eigen-equation cannot be used to realize the spectral representations of infinite dimensions in a finite interval.The method is much simpler and clearer than that of Neumann,and has advantages.The eigen-expansion (eigen-decomposition) of the Hermitian integral operator was given,which has the advantages of theoretical generality and comprehensiveness.The incorrect discussion in Wang et al^[2]was correct that it is used to study the representation of characteristic spectrum.The physical and geometric meanings of naming for the long spherical wave function in optimal eigen-expansion were given.     

Equivalence of generalized joint signal representations ofarbitrary variables

Joint signal representations (JSRs) of arbitrary variables generalize time-frequency representations (TFRs) to a much broader class of nonstationary signal characteristics. Two main distributional approaches to JSRs of arbitrary variables have been proposed by Cohen (see Time-Frequency Analysis, Englewood Cliffs, NJ, Prentice Hall, 1995 and Proc. SPIE 1566, San Diego, 1991) and Baraniuk (see Proc. IEEE Int. Conf. Acoust., Speech Signal Processing, ICASSP'94, vol.3, p.357-60, 1994). Cohen's method is a direct extension of his original formulation of TFRs, and Baraniuk's approach is based on a group theoretic formulation; both use the powerful concept of associating variables with operators. One of the main results of the paper is that despite their apparent differences, the two approaches to generalized JSRs are completely equivalent. Remarkably, the JSRs of the two methods are simply related via axis warping transformations, with the broad implication that JSRs with radically different covariance properties can be generated efficiently from JSRs of Cohen's method via simple pre- and post-processing. The development in this paper, which is illustrated with examples, also illuminates other related issues in the theory of generalized JSRs. In particular, we derive an explicit relationship between the Hermitian operators in Cohen's method and the unitary operators in Baraniuk's approach, thereby establishing the relationship between the two types of operator correspondences     

Difference equation representation of chirp signals andinstantaneous frequency/amplitude estimation

We present a time-varying coefficient difference equation representation for sinusoidal signals with time-varying amplitudes and frequencies. We first obtain a recursive equation for a single chirp signal. Then, using this result, we obtain time-varying coefficient difference equation representations for signals composed of multiple chirp signals. We analyze these equations using the skew-shift operators. We show that the phases of the poles of the difference equations produce instantaneous frequencies (IF), and the magnitudes are proportional to the ratio of successive values of the instantaneous amplitudes (IA). Then algorithms are presented for the estimation of instantaneous frequencies and instantaneous amplitudes for multicomponent signals composed of chirps using the difference equation representation. The first algorithm we propose is based on the skew-shift operators. Next we derive the conditions under which we can use the so-called frozen-time approach. We propose an algorithm for IF and IA estimation based on the frozen-time approach. Then we propose an automatic signal separation method. Finally, we apply the proposed algorithms to single and multicomponent signals and compare the results with some existing methods     

Integral transforms covariant to unitary operators and theirimplications for joint signal representations

Fundamental to the theory of joint signal representations is the idea of associating a variable, such as time or frequency, with an operator, a concept borrowed from quantum mechanics. Each variable can be associated with a Hermitian operator, or equivalently and consistently, as we show, with a parameterized unitary operator. It is well known that the eigenfunctions of the unitary operator define a signal representation which is invariant to the effect of the unitary operator on the signal, and is hence useful when such changes in the signal are to be ignored. However, for detection or estimation of such changes, a signal representation covariant to them is needed. Using well-known results in functional analysis, we show that there always exists a translationally covariant representation; that is, an application of the operator produces a corresponding translation in the representation. This is a generalization of a recent result in which a transform covariant to dilations is presented. Using Stone's theorem, the “covariant” transform naturally leads to the definition of another, unique, dual parameterized unitary operator. This notion of duality, which we make precise, has important implications for joint distributions of arbitrary variables and their interpretation. In particular, joint distributions of dual variables are structurally equivalent to Cohen's class of time-frequency representations, and our development shows that, for two variables, the Hermitian and unitary operator correspondences can be used consistently and interchangeably if and only if the variables are dual     

Unified study on the decomposition for deterministic signals and stationary random signals

The unified decomposition theories and methods for deterministic signals and stationary random signals were deeply studied.According to the stability theory of linear systems,the unified results of signal decomposition under both regular stable and boundary stable conditions were given respectively.The unified results of signal decomposition under both orthogonal projection and self projection conditions were also provided based on the linear space projection theory.The former is clear and definite in its physical meaning,and the latter is clear in its mathematical and geometrical meanings.They are both complement with each other.     

Digital representations of operators on band-limited random signals

Sampling representations of bounded linear operators (BLOs) acting on extended classes of band-limited signals are derived. These are classes of deterministic signals that are not necessarily square-integrable and random signals with covariance functions that are not necessarily square-integrable on the plane. These classes consist of signals whose Fourier transforms, defined as generalized functions (or distributions), have supports that are subsets of compact sets. The merit of these representations lies in the fact that the image of a signal under an operator may be reconstructed from the samples of the original signal rather than the samples of the image. For example, the derivative of a signal, of any order, may be expressed in terms of the samples of the signal itself instead of the samples of the derivative     

A four-parameter atomic decomposition of chirplets

A new four-parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components. The four-parameter chirplet atom is obtained from the unit Gaussian function by successive applications of scaling, fractional Fourier transform (FRFT), and time-shift and frequency-shift operators. The application of the FRFT operator results in a rotation of the Wigner distribution of the Gaussian in the time-frequency plane by a specified angle. The decomposition is realized by using the matching pursuit algorithm. For this purpose, the four-parameter space is discretized to obtain a small but complete subset in the Hilbert space. A time-frequency distribution (TFD) is developed for clear and readable visualization of the signal components. It is observed that the chirplet decomposition and the related TFD provide more compact and precise representation of signal inner structures compared with the commonly used time-frequency representations     

Spherical wave operators and the translation formulas

Translational formulas for both scalar and vector spherical wave solutions of the Helmholtz equation are developed in a straightforward manner using differential operator representations for the modal functions and well-known expressions for the scalar and dyadic free-space Green's functions. The expansion coefficients are given in compact integral or differential operator forms useful for analytic investigation     

谱线辨识方法

谱线辨识是数字信号处理的一个重要课题,利用完全对称窗的实谱特性可以判定分立谱线的存在;相继两次ＤＦＴ分析（实为一次复时域信号分析）可精确求出诸分立谱线的频率、幅度与相位。本文提出的分立谱线辨识方法普遍用于各种周期性时域信号的精密谱分析工作。     

Spectral density of random signals corrupted by multiplicative noise

A simple method of evaluating the spectral density of stationary random signals corrupted by multiplicative noise is presented. It is assumed that the multiplicative noise is also stationary, and that it is statistically independent of the signal.     

Modelling speech signals using formant frequencies as an intermediate representation

Multiple-level segmental hidden Markov models (M-SHMMs) in which the relationship between symbolic and acoustic representations of speech is regulated by a formant-based intermediate representation are considered. New TIMIT phone recognition results are presented, confirming that the theoretical upper-bound on performance is achieved provided that either the intermediate representation or the formant-to-acoustic mapping is sufficiently rich. The way in which M-SHMMs exploit formant-based information is also investigated, using singular value decomposition of the formant-to-acoustic mappings and linear discriminant analysis. The analysis shows that if the intermediate layer contains information which is linearly related to the spectral representation, that information is used in preference to explicit formant frequencies, even though the latter are useful for phone discrimination. In summary, although these results confirm the utility of M-SHMMs for automatic speech recognition, they provide empirical evidence of the value of nonlinear formant-to-acoustic mappings     

Multicomponent AM-FM signal separation and demodulation with null space pursuit

The operator-based signal separation approach, which formulates signal separation as an optimization problem, uses an adaptive operator to separate a signal into additive subcomponents. Furthermore, it is possible to design different operators to fit different signal models. In this paper, we propose a new kind of differential operator to separate multicomponent AM-FM signals. We then use the estimated operators to calculate each sub-component’s envelope and instantaneous frequency. To demonstrate the efficacy of the proposed method, we compare the decomposition and AM-FM demodulation results of several signals, including real-life signals.     

Current and Charge Integral Equation Formulations and Picard's Extended Maxwell System

Important connection between computational and mathematical electromagnetics is presented. The newly developed well-conditioned electromagnetic frequency domain surface integral equation formulations, the current and charge integral equations, are shown to be related to Picard's extended Maxwell system, an extended partial differential equation system that has the correct static behavior. Electromagnetic surface integral representations are derived in this paper for traditional surface integral equation formulations and for the Picard system using the fundamental solution approach, i.e., from the definition of Dirac's delta function. The surface integral representations are constructed with proper solid angle coefficients starting from the scalar Helmholtz equation. The traditional surface integral equation formulations are shown to be derived from Maxwell's curl equations and are thus lacking the contribution of the divergence equations at zero frequency. It is shown that the new current and charge formulations can be derived from the surface integral representation of the Picard system.     

A note on double edge diffraction for parallel wedges

An extended spectral theory of the diffraction approach is used in the investigation of double-edge diffraction in the near-field region of a parallel wedge geometry illuminated by a cylindrical wave. A high-frequency approximation can be formulated in terms of a canonical double stationary phase integral. A uniform asymptotic reduction of the latter, using the method of D.S. Jones (1971), results in a compact representation involving the generalized Fresnel integral, with certain advantages in numerical applications over past representations     

On the cyclostationary statistics of ultra-wideband signals in the presence of timing and frequency jitter

Cyclostationarity is an inherent characteristic of many man-made communication signals, which, if properly recognized, can be exploited for performing various signal-processing tasks. Determining the cyclostationary characteristics of a signal of interest is the first step in the design of signal processing systems exploiting this cyclostationary behaviour. This paper investigates the cyclostationary statistics of various signalling schemes employed in ultra-wideband (UWB) communication systems. Analytical expressions are derived for the cyclic autocorrelation and spectral correlation density functions in the presence of random timing and frequency jitter, which are characterized by discrete-time stationary random processes with known distribution functions.     

Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes

This paper treats estimation of the Wigner-Ville spectrum (WVS) of Gaussian continuous-time stochastic processes using Cohen's class of time-frequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman's sense, and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind. We also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalue-eigenfunction decomposition of a certain Hermitian function. For a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity, the optimal windows are approximately dilated Hermite functions. We determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter     

分数阶Fourier域上非均匀采样信号的频谱重构研究

 本文研究了分数阶Fourier变换域上非均匀采样信号的重构问题.首先得到周期非均匀采样信号经非均匀分数阶Fourier变换后的频谱表达式,研究了该分数阶频谱和信号连续分数阶频谱之间的关系,并基于该关系式提出了一种分数阶Fourier域周期非均匀采样信号的频谱重构算法;其次,讨论了分数阶Fourier变换域上更加一般情况下非均匀采样信号重构问题;最后,给出了周期非均匀采样信号频谱重构的仿真结果.     

基于谱相关的CBOC信号参数盲估计

针对低信噪比下组合二进制偏移载波(CBOC)调制信号的参数盲估计问题,提出了利用谱相关对CBOC信号进行参数估计方法。首先给出了CBOC信号模型,然后根据CBOC信号的数据通道和导频通道之间有良好的正交性特点,详细推导出其谱相关函数可以化简为两个BOC信号谱相关函数的叠加,最后根据CBOC信号循环频率截面的特点进行峰值检索后,实现对伪码速率、载频速率和副载波速率的盲估计。推导结果和计算机仿真分析表明,该方法可以实现在低信噪比下对伪码速率、载频速率和副载波速率的有效估计。     

Kernel decomposition of time-frequency distributions

Bilinear time-frequency distributions (TFDs) offer improved time-frequency resolution over linear representations, but suffer from difficult interpretation, higher implementation cost, and the lack of associated low-cost signal synthesis algorithms. In the paper, the authors introduce some new tools for the interpretation and quantitative comparison of high-resolution TFDs. These tools are used in related work to define low-cost high-resolution TFDs and to define linear, low-cost signal synthesis algorithms associated with high-resolution TFDs. First, each real-valued TFD is associated with a self-adjoint linear operator ψ. The spectral representation of ψ expresses the TFD as a weighted sum of spectrograms (SPs). It is shown that the SP decomposition and Weyl correspondence do not yield useful interpretations for high-resolution TFDs due to the fact that ψ is not positive     

一种多信号环境下的时差估计方法

无线电侦察中的很多信号是周期平稳信号,周期平稳信号的谱相关特性具有选择信号的能力。利用周期平稳信号的谱相关特性,研究了一种进行时差估计的方法,称之为谱相关比值法。研究表明,在多信号环境下,该方法可以准确估计各个来波信号的时差,与传统方法相比,改方法具有较强的抗干扰能力。